Kinetic determination of ultramicro amounts of copper

A new catalytic reaction is proposed and a kinetic method developed for the determination of ultramicro amounts of Cu(II) on the basis of its catalytic activity in oxidation of the 2-thiosemicarbazone of sodium 1,2-naphthoquinone-4-sulphonate by hydrogen peroxide in the presence of ascorbic acid. Under optimal conditions the sensitivity of the method is 0.25 ng/ml. The relative error is 4.8-18.2% for the concentration range 5-0.8 ng/ml. Most foreign ions do not change the rate of the catalysed reaction. Co(2+) and I(-) catalyse the reaction, Ni(2+) extensively inhibits it, and in the presence of EDTA only the uncatalysed reaction takes place.     

Eine kennzeichnende Eigenschaft der Zykel der Galileischen Ebene

Ohne Zusammenfassung     

Effect of the size of experimental channels of the lead slowing-down spectrometer SVZ-100 (Institute for Nuclear Research, Moscow) on the moderation constant

Lead slowing-down (LSD) spectrometers have a low energy resolution (about 30%), but their luminosity is 10³ to 10⁴ times higher than that of time-of-flight (TOF) spectrometers. A high luminosity of LSD spectrometers makes it possible to use them to measure neutron cross section for samples of mass about several micrograms. These features specify a niche for the application of LSD spectrometers in measuring neutron cross sections for elements hardly available in macroscopic amounts—in particular, for actinides. A mathematical simulation of the parameters of SVZ-100 LSD spectrometer of the Institute for Nuclear Research (INR, Moscow) is performed in the present study on the basis of the MCNPX code. It is found that the moderation constant, which is the main parameter of LSD spectrometers, is highly sensitive to the size and shape of detecting volumes in calculations and, hence, to the real size of experimental channels of the LSD spectrometer.     

On general relations in relativistic kinematics and some of their applications

Two vector fields are considered, a timelike one,u , and an arbitrary one, . The relative expansion and rotation are defined with respect to these fields, their mutual relations are studied, and some general formulas obtained. Applications are made, first to vector fields which are mutually nonrotating in a plane-symmetric metric, then to the electromagnetic field of an arbitrary magnetohydrodynamic fluid.     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Numerical solution of a singularly perturbed problem via exponential splines

It is proved that a spline difference scheme for a singularly perturbed self-adjoint problem, derived by using exponential cubic splines at mid-points, has second order uniform convergence in a small parameter . Numerical experiments are presented to confirm the theoretical predictions.     

Some results on starlike trees and sunlike graphs

A tree is called starlike if it has exactly one vertex of degree greater than two. In [4] it was proved that two starlike treesG andH are cospectral if and only if they are isomorphic. We prove here that there exist no two non-isomorphic Laplacian cospectral starlike trees. Further, letG be a simple graph of ordern with vertex setV(G)={1,2, …,n} and letH={H ₁,H ₂, ...H _n } be a family of rooted graphs. According to [2], the rooted productG(H) is the graph obtained by identifying the root ofH _i with thei-th vertex ofG. In particular, ifH is the family of the paths $P_{k_1 } , P_{k_2 } , ..., P_{k_n } $ with the rooted vertices of degree one, in this paper the corresponding graphG(H) is called the sunlike graph and is denoted byG(k ₁,k ₂, …,k _n ). For any (x ₁,x ₂, …,x _n ) ∈I _* ⁿ , whereI _*={0,1}, letG(x ₁,x ₂, …,x _n ) be the subgraph ofG which is obtained by deleting the verticesi ₁, i₂, …,i _j ∈ V(G) (0≤j≤n), provided that $x_{i_1 } = x_{i_2 } = ... = x_{i_j } = 0$ . LetG(x ₁,x ₂,…, x _n] be the characteristic polynomial ofG(x ₁,x ₂,…, x _n ), understanding thatG[0, 0, …, 0] ≡ 1. We prove that $$G[k_1 , k_2 ,..., k_n ] = \Sigma _{x \in ^{I_ * ^n } } \left[ {\Pi _{i = 1}^n P_{k_i + x_i - 2} (\lambda )} \right]( - 1)^{n - (\mathop \Sigma \limits_{i = 1}^n x_i )} G[x_1 , x_2 , ..., x_n ]$$ where x=(x ₁,x ₂,…,x _n );G[k ₁,k ₂,…,k _n ] andP _n (γ) denote the characteristic polynomial ofG(k ₁,k ₂,…,k _n ) andP _n , respectively. Besides, ifG is a graph with λ₁(G)≥1 we show that λ₁(G)≤λ₁(G(k ₁,k ₂, ...,k _n )) < for all positive integersk ₁,k ₂,…,k _n , where λ₁ denotes the largest eigenvalue.     

The Proof That the Standard Transformations of E and B Are Not the Lorentz Transformations

In this paper it is exactly proved that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not relativistically correct transformations. Thence the 3D vectors E and B are not well-defined quantities in the 4D space-time and, contrary to the general belief, the usual Maxwell equations with the 3D E and B are not in agreement with the special relativity. The 4-vectors E ^a and B ^a , as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B. The proof is given in the tensor and the Clifford algebra formalisms.     

Directivity pattern of neurosurgical endoscopic ultrasonic probes

The objective of this paper is to present recent investigations in characteristics of the sound field generated by neurosurgical endoscopic ultrasonic probes (NEUPs) for minimally invasive surgery. The importance of this information has been investigated and discussed taking into account following facts: 1. According to the International Standard IEC 61847:1998 basic acoustical output characteristics of ultrasonic surgical equipment is declared for and measured in an acoustical free field. The standard generally treats the ultrasonic probe as an omnidirectional point source of the zero order (monopole source). 2. In real conditions, operations with NEUPs are performed within the acoustical near field. Having in mind that the cavitational and hydrodynamic effects are dominant, two theoretical boundary conditions can be present. The first one takes place when operations are performed near the "soft" acoustical boundary (tissue/air), and the second one is near the "rigid" acoustical boundary (tissue/bone). Reflections of sound waves from boundaries have influence on the characteristics of the ultrasonic probe (transducer) and on the sound field. In such cases spherical waves of the first and second order are generated. Directivity of sound sources takes shape in the far field and is easier to measure there. On the basis of measured directivity patterns, the influence of different operational conditions (immersion depth of the probe tip, boundary type, acoustical impedance of the medium etc.) on the radiated sound power and spatial distribution of the sound pressure can be estimated.